3 × 1742 = 24 × 177 + 3 × 148 + 3 × 178
3 × 1034 = 12 × 177 + 3 × 148 + 3 × 178
3 × 1210 = 18 × 177 + 3 × 148
---------------------------------------
3 × 3986 = 54 × 177 + 9 × 148 + 6 × 178.
Thus for every 6 parts of 177 days there was consequently 1 of 148 and to
every 9 parts of 177, 1 of 178.
Since 177 and 178 include 6 months each, while 148 equals 5 months, the
entire length of the period is 405 months, which are divided into 69
periods.
It was necessary to discuss all this before I could introduce the entire
series itself. In the following table I have set down the numbers and added
to them the differences between each number and the preceding one (to the
first, the interval between it and the zero point), just as they are given
in the Manuscript. An asterisk is added to show that the number has been
corrected by me and is wrong in the Manuscript, owing to a mistake either
in writing or in computation. The three columns correspond to the three
divisions of 3986 days, and the two horizontal lines divide the periods of
1742, 1034 and 1210 days.
Page 53a: | 24. 4163* 177 | 47. 8149 177
1. 177 177 | 25. 4340 177 | 48. 8326 177
2. 354* 177 | 26. 4488 148* | 49. 8474 148
3. 502 148 | Page 58a: | 50. 8651 177*
4. 679* 177 | 27. 4665 177 | Page 55b:
5. 856 177 | 28. 4842 177 | 51. 8828 177
6. 1034* 178* | 29. 5020 178* | 52. 9006 178*
Page 54a: | 30. 5197 177 | 53. 9183 177
7. 1211 177 | Page 51b: | 54. 9360 177
8. 1388 177 | 31. 5374 177 | 55. 9537 177
9. 1565 177 | 32. 5551 177 | 56. 9714 177
10. 1742* 177 | 33. 5728 177 |
-----------------+-------------------+-----------------
11. 1919 177 | 34. 5905 177 | 57. 9891 177
12. 2096* 177 | 35. 6082 177 | 58. 10068* 177*
13. 2244* 148 | 36. 6230 148 | Page 56b:
Page 55a: | Page 52b: | 59. 10216 148*
14. 2422* 178 | 37. 6408 178* | 60. 10394 178*
15. 2599* 177 | 38. 6585 177 | 61. 10571 177
16. 2776 177 | 39. 6762 177 | 62. 10748 177
-----------------+-------------------+-----------------
17. 2953 177 | 40. 6939 177 | Page 57b:
18. 3130 177 | Page 53b: | 63. 10925 177
Page 56a: | 41. 7116 177 | 64. 11102 177
19. 3278 148 | 42. 7264 148 | 65. 11250 148
20. 3455 177 | 43. 7441 177 | 66. 11427 177
21. 3632 177 | 44. 7618 177 | 67. 11604 177
22. 3809 177 | 45. 7795 177 | Page 58b:
Page 57a: | Page 54b: | 68. 11781 177
23. 3986 177* | 46. 7972 177 | 69. 11958 177
No one acquainted with the cursoriness of the Maya Manuscripts will be
surprised that among 138 numbers I have declared 21 to be wrong.
Furthermore the 21 errors are lessened by the fact that six of them are
really only one, for in all 6 cases where the difference is 178, the scribe
has overlooked this and written down the usual 177, although the numbers
and the days of the series very correctly indicate 178. Again the three
errors in groups 58 and 59 are also only one, for the author had confused
the differences 177 and 148 and had, therefore, to write down 10,039
instead of 10,068. In group 4 the error is merely the omission of a line
meaning 5. The scribe must have been at the same time the computer and
therefore the actual author of the Manuscript.
Furthermore I must call attention to the regular position of the
differences 178 and 148. In the three periods of 1742 days the 178 always
occupies the 6th place and in the periods of 1034 it is always in the 4th
place. This difference appears, therefore, in groups 6, 14, 29, 37, 52 and
60, _i.e._, 8, 15, 8, 15 and 8 groups apart; but it is entirely lacking in
the periods of 1210 days. And in all nine sections the 148 occupies the
third place, _i.e._, directly in front of the pictures, which will be
discussed immediately, therefore in groups 3, 13, 19, 26, 36, 42, 49, 59,
65, _i.e._, at intervals of 10, 6, 7, 10, 6, 7, 10 and 6 groups. But I must
point out an error fraught with consequences. Groups 22 and 23 quite
correctly have the difference 177, but in this single instance the scribe
has written down 178 and hence has computed the three days belonging to it
as VII Ix, VIII Men and IX Cib instead of VI Ben, VII Ix and VIII Men, and
from here on to the close he is always one day in advance, so that on page
58 group 69 ends with the days X Cimi, XI Manik and XII Lamat, while it
ought to have ended with IX Chicchan, X Cimi and XI Manik.
So much for the series. Vid. on this series my paper "Zwei
Hieroglyphenreihen in der Dresdener Mayahandschrift" (Zeitschrift für
Ethnologie, 1905, numbers 2 and 3). Let us turn next to the ten pictures
which are inserted in this series, three of which appear at the end of each
period of 2920 days as on pages 46-50. Let us attempt to advance a step
further in the darkness which still surrounds us here.
One of these pictures, the 8th, which is on page 56b, is in the wrong
place, owing to the error in computation in Groups 58 and 59 to which I
called attention above. It belongs not _before_ but _after_ group 59, the
first on page 56b. This is indicated in the Manuscript itself. For in group
59 the two hieroglyphs, usually placed above each group, are missing and we
find instead of them the sign resembling a snail, which is doubtless a very
much emphasized zero (compare my "Erläuterungen," page 29), which indicates
that the section designated by a picture closes with this group.
Having corrected this error we see that the ten pictures are on the
following pages and come after the following numbers of the series:--
1. 53a 502
2. 55a 2244
3. 56a 3278
4. 57a 4488
5. 52b 6230
6. 53b 7264
7. 54b 8474
8. 56b 10216
9. 57b 11250
10. 58b 11958.
From this it follows that a picture is assigned to each of the nine
sections composing the series. They are placed, however, not at the
beginning or end of the section, but always after the expiration of 502 (=
2 × 177 + 148) days. The pictures are thus separated from one another by
1742, 1034 and 1210 days, which intervals correspond exactly to the length
of the nine sections. But the last picture is separated from the preceding
one by 708 days, and as it has a character quite its own, it must be
discussed separately. But these 708 days with the 502 days of the beginning
quite regularly amount to 1210 days, and the series is therefore to be
considered as a recurring one.
Now these nine pictures might very easily be regarded as forming a new
series, which is inserted in the original one and which has the day 502 as
its zero-point. In that case, we shall have to subtract 502 every time from
the days set down in the Manuscript. This new series may be represented
thus:--
1. 53a 0
2. 55a 1742
3. 56a 2776
4. 57a 3986
5. 52b 5728
6. 53b 6762
7. 54b 7972
8. 56b 9714
9. 57b 10748.
It is certainly remarkable that the last number, 10748, corresponds so
closely to the time of the revolution of Saturn, which is computed at 10753
days. For owing to the slowness of its progress, the Mayas may have known
not only the apparent but also the actual revolution of Saturn. Besides the
apparent revolution of Saturn (378 days from one superior conjunction to
the next) could not be made to coincide very well with the length of the
solar year. I will immediately present a further confirmation of my theory.
All these pictures have rectangles above them, of which I have spoken in my
"Erläuterungen," page 16, and which always enclosed two or three
hieroglyphs in which, with due hesitation, I assumed to be the signs of the
sun, moon, and planets. This theory has as yet called forth no serious
opposition.
Now in the passages just mentioned, I indicated the following figures as
the signs of Saturn:--
[Illustration]
These figures actually occur in all the nine pictures with the exception of
the first, which has no rectangle at all, and where in true Maya fashion,
the zero-point is concealed.
I go still further in my bold hypothesis. The time of the apparent
revolution of Jupiter has been placed at 397 days. The Mayas, I think,
computed it at 398 days. In the passage alluded to I regarded the following
as the sign for Jupiter:--
[Illustration]
We find these signs in pictures 4, 6, 7 and 9. The corresponding numbers
reduced for the revolution of Saturn are 3986, 6762, 7972 and 10,748. I
assume that the third picture, _i.e._, the number 2776, is another
zero-point, in consequence of which the sign is here suppressed, and that
still another is the tenth picture with the number 11,958, which has no
relation to the revolution of Saturn.
If we compare these numbers with the 398, _i.e._, the apparent revolution
of Jupiter, we have the following:--
3. 2776 = 7 × 398 - 10
4. 3986 = 10 × 398 + 6
6. 6762 = 17 × 398 - 4
7. 7972 = 20 × 398 + 12
9. 10748 = 27 × 398 + 2
10. 11958 = 30 × 398 + 18
The differences 10, 6, 4, 12, 2 and 18 are so small in comparison with 398,
that the numbers 2776, etc., might very well have been regarded as
approximate multiples of the revolution of Jupiter. And the remainders in
the seventh and tenth pictures could be still further reduced. In the
seventh picture, the first sign is very unusual and one which I do not
remember having met with elsewhere. If it should be possible to regard it
as the number of the thirteen week days, then it would follow (the Saturn
sign being regarded as unimportant) that the contents of the rectangle
meant:-- 13 + a multiple of 398, by which this remainder would be reduced
to -1.
The tenth picture has the cross _b_ as the beginning of the rectangle. This
is the sign for union, very often denoting especially the union of all the
twenty days. Thus we have here (aside from the middle sign to be discussed
later) the formula:-- 20 + 30 × 398 - 2 = 11,958, or even 20 + 30 × 398 =
11,960.
The regular progression from the 7th multiple to the 10th, 17th, 20th,
27th, and 30th multiples in the above six equations is also somewhat in
favor of my theory, while the four rectangles without the Jupiter sign are
by no means multiples of the Jupiter revolution:--
1. 502 = 398 + 104
2. 1742 = 4 × 398 + 150
5. 5728 = 14 × 398 + 156
8. 9714 = 24 × 398 + 162.
Let us now try to interpret the meaning of the remaining rectangles (always
omitting the Saturn sign as a matter of course.)
In pictures 2 and 8 the rectangle also contains the sign of the moon or of
the twenty days. Beside it in picture 2 is the sign, which in my
"Erläuterungen," page 16, I regarded as the sign for Mercury. Hence we have
here 20 + 15 × 115 = 1745, _i.e._, only 3 units more than the required
1742.
The rectangle with the eighth picture contains in addition to the moon a
sign which looks as if it were intended for a whole divided into four
parts. Until something better (perhaps the the sign of Venus) is proposed,
I will assume that it is the quarter of the Tonalamatl, _i.e._, 65, and I
take the required number to be 9714 in the form of 20 + 149 × 65 + 9.
Above the third picture I see a Mercury and a Venus sign and I read 584 +
19 × 115 = 2769, which is only 7 units less than the required 2776.
The fifth picture still remains to be discussed, but I do not know how to
unite the Mercury revolution here with the 5728. For the present, however,
I am inclined to believe that there is a mistake in this passage.
We pass now from the obscure contents of the rectangles to the equally
mysterious pictures themselves.
Aside from the tenth picture, I find human forms in four pictures.
Picture 1, page 53a, is the death-god (A) seated and pointing upward, an
appropriate representation for the zero-point of the Saturn series, _i.e._,
for the end of the preceding revolution.
Picture 2, page 55a, contains the head of a deity, probably D's with the
suggestion of a beard and the sun-sign on his forehead. The head is
surrounded by a ring striped black and white.
Picture 3, page 56a, is the head of B, again with a beard and with the sign
Kin (sun) above. The head is surrounded by a design, the left part of which
is black and the right white.
Picture 6, page 53b, represents a hanged woman, which Schellhas,
"Göttergestalten," page 11, takes to be the Maya goddess Ixtab, the goddess
of the halter, _i.e._, of the hanged.
The centre of picture 4 on page 57a, contains the suggestion of a face,
perhaps in place of the Ahau sign, and on either side of it is a black and
white surface.
It is further important to note that four times in this section Kin (sun)
forms the centre of the picture, viz:--pictures 5, 7, 8 and 9, pages 52b,
54b, 56b and 57b. In all four cases there is on either side of Kin a black
and white surface, such as we have already seen in picture 4 and similar to
that in picture 3. Pictures 8 and 9 are vomited up, as it were, by a
serpent placed below them, in the same way as B is represented on pages 34b
and 35b. In pictures 5 and 8, four objects suggesting arrows extend from
the Kin in four directions and probably denote the four cardinal points or
the four Bacabs, of which we shall have more to say presently. Two of these
arrow-like signs also appear in picture 7, page 54b, but only on the black
and not on the white surface.
I will postpone discussing picture 10 until later and pass on to the
hieroglyphs above the first nine pictures, about which it is true I have
nothing satisfactory to say. There are always properly speaking ten of
these hieroglyphs, among them the two signs for the sun and moon. But the
scribe introduced the latter only in pictures 1-4, and also with the more
elaborate last picture 10. With pictures 5 and 9 he omitted these signs in
order to represent the other eight larger and with greater distinctness of
detail. Among these hieroglyphs are several of gods, especially that of A
with pictures 1, 5 and 9, and H with picture 5, and with pictures 1, 3, 5,
7, 8 and 9 there are other heads, some of them bird-heads, regarding which
I am uncertain.
The Ben-Ik sign, to which I have assigned the meaning of a lunar month,
belongs with pictures 4, 8 and 9 and occurs twice each with pictures 1 and
10.
I am inclined to see the sign for Mercury in the crouching figure belonging
to pictures 9 and 10, which is drawn upside down and combined with the half
Venus sign (11958 = 104 × 115 - 2).
Hands grasping a hieroglyph (a sign for 20 days?) are represented in
pictures 1, 7, 8 and 10.
The enigmatical numbers, prefixed to the hieroglyphs, occur several times,
thus a 1 with pictures 1 and 10, and a 4 twice with picture 8 and a 6 with
picture 3.
Now let us examine picture 10 somewhat in detail and also the signs
standing above it, since both are of special significance here. This
representation treats of the period of 11,960 days in which the Mercury and
lunar revolutions meet. And this is proved by the ten hieroglyphs, which I
will number as follows:--
1 6
2 7
3 8
4 9
5 10.
I can omit Signs 3 and 8, sun and moon, since they refer to a period of
time only in a general way. Sign 1 seems to me, as I have already stated,
to have reference to the revolutions of Mercury. Then follows sign 2, the
upper part of which is a mat and the lower the Muluc sign. I believe this
sign is intended to denote that the beginning of this period is in a Muluc
year. Indeed, our examination of pages 51-52 showed that it was the year 6
Muluc. The mat (Pop) is very properly the symbol of beginning, since the
first month of the year was likewise called Pop. Sign 7, it seems to me,
indicates that this period should be divided into lunar months (denoted by
Ben-Ik), and, as I have already demonstrated in my examination of page 24,
the length of the period is stated here by Signs 4, 5, 6 and 9, but the dot
before the fifth should be placed before the fourth, as is actually the
case on page 24. Therefore:--
4 = 21
5 = 7200
6 = 4680
9 = 59
-----
11960.
It is perhaps not accidental that the ninth sign is that of the fourteenth
month, which signifies the expiration of the preceding lunar month, for
here the month begins with the first day of the fifteenth month.
Sign 10 is doubtless Xul = end, as it so often is, for example, on pages
61-62 below. But I have not solved the meaning of the two prefixes. The end
would be XII Lamat 16 Yax (13 Ix).
The picture represents a human form, which has in place of a head a design
somewhat resembling the head of a lance. It is sitting with legs spread
apart, and in this respect may be compared with god B of Cort. 9, who is
represented in the same way. In the picture before us, the figure holds in
its upraised hands the sun and moon signs, which are constantly repeated
throughout the series. The Venus sign is placed between the outspread legs.
In the rectangle above the figure, this sign is repeated in a more concise
form, while on the left the cross _b_ appears as the sign of union or
multiplication, and on the right that of Jupiter, whose period of
revolution is here multiplied by 30 (30 × 398 = 11,940). And the two Venus
signs can mean nothing more than that this period of 11,960 also serves the
purpose of filling up the gap between the two large Venus-solar periods of
37,960 days, like the similar process which we saw on pages 46-50.
We have examined first the series and then the pictures with the
hieroglyphs belonging to them. Let us pass now, as the third step, to the
examination of the two rows of hieroglyphs extending above the numbers
throughout the whole section. First of all, I will again set down here the
position of each of the sixty-nine groups:--
Page 51. | 52. | 53. | 54.
31.32.33.34.35.36.|37.38.39.40.| 1. 2. 3. 4. 5. 6.| 7. 8. 9.10.11.12.13.
| |41.42. 43.44. 45. |46.47.48.49.50.
Page 55. | 56. | 57. | 58.
14.15.16.17.18. |19.20.21.22.|23.24.25.26. |27.28.29.30.
51.52.53.54.56.57.58.|59.60.61.62.|63.64.65.66.67.|68.69.
Since each group contains two hieroglyphs, this makes 138. in all. Of
these, however, about 24 on the upper halves of the pages, are wholly or
almost wholly effaced which very materially hinders the trustworthy
determination of the context.
Furthermore group 59 is entirely lacking or rather group 58, in the place
of which the 59th has been set down. The eighth picture was probably
already drawn, when the artist saw that there was not room enough left for
the 58th and 59th groups. Hence he omitted the 58th, setting down in place
of it the 59th and in the place of the latter he set down the zero
mentioned above.
The question now arises:--Are these hieroglyphs dependent upon the days and
numbers of the series and upon the pictures, or are they entirely
independent of them?
I find but _one_ point in favor of the first possibility, viz:--the Venus
sign in group 4b (I will designate the upper hieroglyphs by _a_ and the
lower by _b_). It is placed in the period indicated in which 502-679 days
elapse, and in which, therefore, Venus has finished a revolution of 584
days. It may be, that by way of exception, this significant date was
intentionally recorded.
On the other hand, there are many things, which favor an entirely different
interpretation of these hieroglyphs. Thus I am of the opinion that the
ritual year of 364 days with its four Bacab periods of 91 days each is
referred to here, as we have already found it referred to on pages 31a-32a
and on page 45a, and shall find it again on pages 65-69 and 71-73. In that
case the single groups would be separated from one another by one Maya week
= 13 days.
I will now arrange the sixty-nine groups in the following order (the reason
for which will become clear directly):--
I 4 11 18 25 32 39 46 53 60 67
II 5 12 19 26 33 40 47 54 61 68
III 6 13 20 27 34 41 48 55 62 69
IV 7 14 21 28 35 42 49 56 63 1
V 8 15 22 29 36 43 50 57 64 2
VI 9 16 23 30 37 44 51 58 65 3
VII 10 17 24 31 38 45 52 59 66.
The groups in a horizontal row are separated from one another by 7 or a
multiple of 7. If now a hieroglyph is repeated in those places, which are
in the same horizontal row, then this is a confirmation of the supposition
that Bacab periods are meant to be represented here. Hence I will examine
each row in turn. These rows extend over the long period of 69 × 13 days
probably merely for the purpose of filling up the space.
I. In 39b, 46b, 53b and 60b, _i.e._, after every seven groups, perhaps also
in 18b, we find the following sign, which I identified as that of a Bacab,
in Globus, Vol. LXXI:--
[Illustration]
Hence this denotes the beginning of the Bacab period. In 4b the sign is
replaced by that for Venus. In 11b, 25b, 32b and 67b we find other signs,
it is true, nevertheless the regularity stated above cannot be accidental.
The upper signs of groups 39a, 46a, 53a and 60a contain an Imix and
corroborate the connection.
II. 5b and 26b (after 3 × 91 days) contain a head very like the preceding,
which readily suggests the idea that it is merely a Bacab sign pushed one
group ahead, but it also appears in 13b, 50b and 52b.
Then 12b, 54b and 61b correspond, _i.e._, after six groups of 91 days and
one more of the same length, but the same sign appears also in 34b, 48b and
56b.
III. 41b and 69a are Xul = end and are therefore separated by 28 × 13 = 4 ×
91 days, _i.e._, the length of a year. It is singular that both signs of 41
are like those of 47; if we assume that 47 was set down one group too soon,
it would be in excellent keeping with the rest. The Xul also appears in 11b
and 28b. 34b and 48b correspond after 2 × 91 days, as already mentioned
under II.